Problem 1

we want to find \(\int_0^\infty \log(1+x^2)e^{-x}dx\).

Use simple simulation and antithetic variables. What are their respective variances?

Problem 2

Let \(X_1,..,X_{10}\sim Pois(1)\) and independent. Find \(P(\max\{X_i\}>10)\)

  1. analytically (you can use the R command ppois)

  2. via importance sampling